Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom.
xi is the estimated shape parameter.

Log returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of gross returns

##       vmr             pmr             mmr             vhr       
##  Min.   :0.868   Min.   :0.904   Min.   :0.988   Min.   :0.849  
##  1st Qu.:1.044   1st Qu.:1.042   1st Qu.:1.013   1st Qu.:1.039  
##  Median :1.097   Median :1.084   Median :1.085   Median :1.099  
##  Mean   :1.070   Mean   :1.065   Mean   :1.066   Mean   :1.085  
##  3rd Qu.:1.136   3rd Qu.:1.107   3rd Qu.:1.101   3rd Qu.:1.160  
##  Max.   :1.168   Max.   :1.141   Max.   :1.133   Max.   :1.214  
##       phr             mhr       
##  Min.   :0.878   Min.   :0.977  
##  1st Qu.:1.068   1st Qu.:1.013  
##  Median :1.128   Median :1.113  
##  Mean   :1.095   Mean   :1.087  
##  3rd Qu.:1.182   3rd Qu.:1.128  
##  Max.   :1.208   Max.   :1.207
##       vmrl      
##  Min.   :0.801  
##  1st Qu.:1.013  
##  Median :1.085  
##  Mean   :1.061  
##  3rd Qu.:1.128  
##  Max.   :1.193
## Highest minimum log-return: mmr
## Highest median log-return: phr
## Highest mean log-return: phr
## Highest max log-return: vhr
## cov(vmr, pmr) =  -0.001094875
## cov(vhr, phr) =  -0.0001730651

Velliv medium risk, 2011 - 2023

## 
## AIC: -27.8497 
## BIC: -25.58991 
## m: 0.0480931 
## s: 0.1198426 
## nu (df): 3.303595 
## xi: 0.03361192 
## R^2: 0.993 
## 
## An R^2 of 0.993 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 7.4 percent
## What is the risk of losing max 25 %? =< 1.8 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 41 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 280.482 kr.
## SD of portfolio index value after 20 years: 125.271 kr.
## Min total portfolio index value after 20 years: 4.875 kr.
## Max total portfolio index value after 20 years: 1015.984 kr.
## 
## Share of paths finishing below 100: 5 percent

Velliv medium risk, 2007 - 2023

Fit to skew t distribution

## 
## AIC: -34.35752 
## BIC: -31.02467 
## m: 0.05171176 
## s: 0.1149408 
## nu (df): 2.706099 
## xi: 0.5049945 
## R^2: 0.978 
## 
## An R^2 of 0.978 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.4 percent
## What is the risk of losing max 25 %? =< 1.3 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 36.2 percent
## What is the chance of gaining min 25 %? >= 0.3 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 295.167 kr.
## SD of portfolio index value after 20 years: 143.102 kr.
## Min total portfolio index value after 20 years: 0.528 kr.
## Max total portfolio index value after 20 years: 8328.697 kr.
## 
## Share of paths finishing below 100: 3.52 percent

Velliv high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -21.42488 
## BIC: -19.16508 
## m: 0.06471454 
## s: 0.1499924 
## nu (df): 3.144355 
## xi: 0.002367034 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 8.3 percent
## What is the risk of losing max 25 %? =< 2.5 percent
## What is the risk of losing max 50 %? =< 0.4 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 53.3 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 405.997 kr.
## SD of portfolio index value after 20 years: 218.082 kr.
## Min total portfolio index value after 20 years: 0.015 kr.
## Max total portfolio index value after 20 years: 1613.514 kr.
## 
## Share of paths finishing below 100: 4.28 percent

PFA medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -33.22998 
## BIC: -30.97018 
## m: 0.05789224 
## s: 0.1234592 
## nu (df): 2.265273 
## xi: 0.477324 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.9 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 32.7 percent
## What is the chance of gaining min 25 %? >= 0.1 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.

##  [1] -0.091256521 -0.003731241  0.027312079  0.045808232  0.059068633
##  [6]  0.069575113  0.078454727  0.086316936  0.093536451  0.100370932
## [11]  0.107018607  0.114081432  0.127604387

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 322.507 kr.
## SD of portfolio index value after 20 years: 103.889 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1373.96 kr.
## 
## Share of paths finishing below 100: 1.95 percent

PFA high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -23.72565 
## BIC: -21.46585 
## m: 0.08386034 
## s: 0.1210107 
## nu (df): 3.184569 
## xi: 0.01790306 
## R^2: 0.964 
## 
## An R^2 of 0.964 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.3 percent
## What is the risk of losing max 25 %? =< 1.4 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 59.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 552.944 kr.
## SD of portfolio index value after 20 years: 242.594 kr.
## Min total portfolio index value after 20 years: 0.239 kr.
## Max total portfolio index value after 20 years: 1691.442 kr.
## 
## Share of paths finishing below 100: 1.14 percent

Mix medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -36.9603 
## BIC: -34.7005 
## m: 0.05902873 
## s: 0.08757749 
## nu (df): 2.772621 
## xi: 0.02904471 
## R^2: 0.89 
## 
## An R^2 of 0.89 suggests that the fit is not completely random.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.7 percent
## What is the risk of losing max 50 %? =< 0.1 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 35.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The fit suggests big losses for the lowest percentiles, which are not present in the data.
So the fit is actually a very cautious estimate.

Data vs fit

Let’s plot the fit and the observed returns together.

Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 324.451 kr.
## SD of portfolio index value after 20 years: 99.353 kr.
## Min total portfolio index value after 20 years: 0.001 kr.
## Max total portfolio index value after 20 years: 671.691 kr.
## 
## Share of paths finishing below 100: 1.28 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 300.888 kr.
## SD of portfolio index value after 20 years: 86.98 kr.
## Min total portfolio index value after 20 years: 49.73 kr.
## Max total portfolio index value after 20 years: 3323.422 kr.
## 
## Share of paths finishing below 100: 0.33 percent

Mix high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -24.26084 
## BIC: -22.00104 
## m: 0.0822419 
## s: 0.07129843 
## nu (df): 89.86289 
## xi: 0.7697502 
## R^2: 0.961 
## 
## An R^2 of 0.961 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0.9 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 46.1 percent
## What is the chance of gaining min 25 %? >= 1.2 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that the high risk mix provides a much better upside and smaller downside.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 502.41 kr.
## SD of portfolio index value after 20 years: 156.682 kr.
## Min total portfolio index value after 20 years: 135.603 kr.
## Max total portfolio index value after 20 years: 1580.187 kr.
## 
## Share of paths finishing below 100: 0 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 478.578 kr.
## SD of portfolio index value after 20 years: 161.465 kr.
## Min total portfolio index value after 20 years: 59.4 kr.
## Max total portfolio index value after 20 years: 1264.798 kr.
## 
## Share of paths finishing below 100: 0.14 percent

Compare pension plans

Risk of max loss of x percent for a single period (year).
x values are row names.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
0 21.3 18.2 19.9 12.2 14.3 12.7 13.0
5 12.5 9.6 12.8 6.0 8.6 6.2 4.2
10 7.4 5.4 8.3 3.3 5.3 3.3 0.9
25 1.8 1.3 2.5 0.9 1.4 0.7 0.0
50 0.2 0.2 0.4 0.2 0.2 0.1 0.0
90 0.0 0.0 0.0 0.0 0.0 0.0 0.0
99 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Chance of min gains of x percent for a single period (year).
x values are row names.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
0 78.7 81.8 80.1 87.8 85.7 87.3 87.0
5 63.8 64.9 69.2 71.5 75.8 71.4 69.9
10 41.0 36.2 53.3 32.7 59.6 35.6 46.1
25 0.0 0.3 0.0 0.1 0.0 0.0 1.2
50 0.0 0.0 0.0 0.0 0.0 0.0 0.0
100 0.0 0.0 0.0 0.0 0.0 0.0 0.0

MC risk percentiles: Risk of loss from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

_m is medium.
_h is high.

Velliv_m Velliv_m_long Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 5.00 3.52 4.28 1.95 1.14 1.28 0 0.33 0.14
5 4.33 3.11 3.92 1.79 1.02 1.13 0 0.26 0.09
10 3.56 2.83 3.47 1.59 0.92 0.95 0 0.20 0.07
25 2.13 1.95 2.22 1.15 0.68 0.70 0 0.10 0.04
50 0.60 0.89 1.15 0.53 0.40 0.25 0 0.01 0.00
90 0.04 0.11 0.14 0.09 0.08 0.03 0 0.00 0.00
99 0.00 0.02 0.07 0.03 0.01 0.01 0 0.00 0.00

MC gains percentiles: Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

Velliv_m Velliv_m_long Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 95.00 96.48 95.72 98.05 98.86 98.72 100.00 99.67 99.86
5 94.30 95.99 95.22 97.96 98.73 98.52 100.00 99.56 99.85
10 93.60 95.57 94.85 97.79 98.62 98.39 100.00 99.47 99.81
25 91.10 93.86 93.44 96.88 98.09 97.69 100.00 99.04 99.66
50 85.90 90.11 90.60 94.92 97.24 95.97 99.98 97.84 99.22
100 71.75 78.44 83.02 88.42 94.72 89.36 99.67 89.66 97.58
200 39.42 45.33 64.53 59.06 85.66 59.86 93.47 48.44 87.48
300 16.53 17.57 45.08 22.44 71.29 22.88 72.46 10.98 66.46
400 5.57 5.01 29.27 3.87 54.66 3.77 44.67 1.28 41.46
500 1.49 1.17 17.49 0.50 38.73 0.26 23.82 0.15 21.20
1000 0.00 0.02 0.67 0.01 2.12 0.00 0.21 0.01 0.08

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
m 0.048 0.052 0.065 0.058 0.084 0.059 0.082
s 0.120 0.115 0.150 0.123 0.121 0.088 0.071
nu 3.304 2.706 3.144 2.265 3.185 2.773 89.863
xi 0.034 0.505 0.002 0.477 0.018 0.029 0.770
R-squared 0.993 0.978 0.991 0.991 0.964 0.890 0.961

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
Velliv_m Velliv_m_long Velliv_h PFA_m PFA_h mix_m_a mix_m_b mix_h_a mix_h_b
mc_m 280.482 295.167 405.997 322.507 552.944 324.451 300.888 502.410 478.578
mc_s 125.271 143.102 218.082 103.889 242.594 99.353 86.980 156.682 161.465
mc_min 4.875 0.528 0.015 0.000 0.239 0.001 49.730 135.603 59.400
mc_max 1015.984 8328.697 1613.514 1373.960 1691.442 671.691 3323.422 1580.187 1264.798
dao_prob_pct 0.000 0.000 0.000 0.010 0.000 0.000 0.000 0.000 0.000
losing_prob_pct 5.000 3.520 4.280 1.950 1.140 1.280 0.330 0.000 0.140 
## Highest mean    : PFA_h
## Lowest sd       : mix_m_b
## Highest min     : mix_h_a
## Highest max     : Velliv_m_long
## Lowest dao prob : Velliv_m Velliv_m_long Velliv_h PFA_h mix_m_a mix_m_b mix_h_a mix_h_b
## Lowest loss prob: mix_h_a

Appendix

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t y_{t-1} + x_{t-1} y_t }{x_{t-1} y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}} \] \[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

We are adding annual returns rather than multiplying, so imagine that we are simulating log returns.

## m(data_x): 0.1021886 
## s(data_x): 0.4359335 
## m(data_y): 9.509825 
## s(data_y): 3.139521 
## 
## m(data_x + data_y): 4.806007 
## s(data_x + data_y): 1.580826

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
95.850 96.133 7.109 6.974
96.059 96.501 7.225 7.053
95.994 96.246 7.251 6.988
95.933 96.150 7.252 7.173
96.170 96.227 7.167 7.130
95.939 95.831 7.309 6.987
96.365 96.033 7.064 6.919
96.294 95.889 7.204 7.210
96.185 96.026 7.235 7.040
96.509 96.102 7.080 7.059 
##       m_a             m_b             s_a             s_b       
##  Min.   :95.85   Min.   :95.83   Min.   :7.064   Min.   :6.919  
##  1st Qu.:95.95   1st Qu.:96.03   1st Qu.:7.123   1st Qu.:6.987  
##  Median :96.11   Median :96.12   Median :7.215   Median :7.047  
##  Mean   :96.13   Mean   :96.11   Mean   :7.190   Mean   :7.053  
##  3rd Qu.:96.27   3rd Qu.:96.21   3rd Qu.:7.247   3rd Qu.:7.112  
##  Max.   :96.51   Max.   :96.50   Max.   :7.309   Max.   :7.210

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

## [1] 0.005355618
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05968   Min.   :0.04221  
##  1st Qu.:0.06610   1st Qu.:0.05953  
##  Median :0.07093   Median :0.06847  
##  Mean   :0.07135   Mean   :0.06810  
##  3rd Qu.:0.07602   3rd Qu.:0.07439  
##  Max.   :0.08455   Max.   :0.08938